Theorem tpssd 32921

Description: Deduction version of tpssi : An unordered triple of elements of a class is a subset of that class. (Contributed by Thierry Arnoux, 2-Nov-2025.)

Hypotheses

Ref	Expression
tpssd.1	⊢ (𝜑 → 𝐴 ∈ 𝐷)
tpssd.2	⊢ (𝜑 → 𝐵 ∈ 𝐷)
tpssd.3	⊢ (𝜑 → 𝐶 ∈ 𝐷)

Assertion

Ref	Expression
tpssd	⊢ (𝜑 → {𝐴, 𝐵, 𝐶} ⊆ 𝐷)

Proof of Theorem tpssd

Step	Hyp	Ref	Expression
1		tpssd.1	. 2 ⊢ (𝜑 → 𝐴 ∈ 𝐷)
2		tpssd.2	. 2 ⊢ (𝜑 → 𝐵 ∈ 𝐷)
3		tpssd.3	. 2 ⊢ (𝜑 → 𝐶 ∈ 𝐷)
4		tpssi 4808	. 2 ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐷 ∧ 𝐶 ∈ 𝐷) → {𝐴, 𝐵, 𝐶} ⊆ 𝐷)
5	1, 2, 3, 4	syl3anc 1398	1 ⊢ (𝜑 → {𝐴, 𝐵, 𝐶} ⊆ 𝐷)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2146   ⊆ wss 3908  {ctp 4598

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2738

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2745  df-cleq 2758  df-clel 2841  df-v 3460  df-un 3913  df-ss 3925  df-sn 4595  df-pr 4597  df-tp 4599

This theorem is used by:  constrlccllem  34174  constrcccllem  34175  cos9thpiminplylem2  34204